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Issue ESAIM: M2AN
Volume 43, Number 5, September-October 2009
Page(s) 929 - 955
DOI 10.1051/m2an/2009013
Published online 12 June 2009

ESAIM: M2AN 43 (2009) 929-955
DOI: 10.1051/m2an/2009013

A convergence result for finite volume schemes on Riemannian manifolds

Jan Giesselmann

Universität Stuttgart (IANS), Pfaffenwaldring 57, 70569 Stuttgart, Germany.  jan.giesselmann@mathematik.uni-stuttgart.de

Received July 30, 2008. Revised November 24, 2008. Published online June 12, 2009.

Abstract
This paper studies a family of finite volume schemes for the hyperbolic scalar conservation law $u_t +\nabla_g \cdot f(x,u)=0$ on a closed Riemannian manifold M. For an initial value in BV(M) we will show that these schemes converge with a $h^{\frac{1}{4}} $ convergence rate towards the entropy solution. When M is 1-dimensional the schemes are TVD and we will show that this improves the convergence rate to $h^{\frac{1}{2}}.$


Mathematics Subject Classification. 74S10, 35L65, 58J45

Key words: Finite volume method, conservation law, curved manifold


© EDP Sciences, SMAI 2009


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